By the defining property of tensoring and cotensoring (or explicitly writing out S⋅y(U)=lim→Sconsty(U)S \cdot y(U) = {\lim_\to}_{S} const y(U) , taking the colimit out of the hom, thus turning it into a limit and then inserting that back in the second argument) this is

Moreover, it turns out that (∞,1)(\infty,1)-toposes come with plenty of internal structures, more than canonically present in an ordinary topos. Every (∞,1)(\infty,1)-topos comes with its intrinsic notion of

building on ideas by Charles Rezk. There is is also proven that the Brown-Joyal-Jardine-Toën-Vezzosi models indeed precisely model ∞\infty-stack (∞,1)(\infty,1)-toposes. Details on this relation are at models for ∞-stack (∞,1)-toposes.